Lemma 79.12.3. Denote the common underlying category of $\mathit{Sch}_{fppf}$ and $\mathit{Sch}_{\acute{e}tale}$ by $\mathit{Sch}_\alpha$ (see Topologies, Remark 34.11.1). Let $S$ be an object of $\mathit{Sch}_\alpha$.

$F : (\mathit{Sch}_\alpha /S)^{opp} \longrightarrow \textit{Sets}$

be a presheaf with the following properties:

1. $F$ is a sheaf for the étale topology,

2. the diagonal $\Delta : F \to F \times F$ is representable by algebraic spaces, and

3. there exists $U \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha /S)$ and $U \to F$ which is surjective and smooth.

Then $F$ is an algebraic space in the sense of Algebraic Spaces, Definition 64.6.1.

Proof. The proof mirrors the proof of Lemma 79.12.1. Let $R = U \times _ F U$. By (2) the presheaf $R$ is an algebraic space and by (3) the projections $R \to U$ are smooth and surjective. Denote $(U, R, s, t, c)$ the groupoid associated to the equivalence relation $j : R \to U \times _ S U$ (see Groupoids in Spaces, Lemma 77.11.3). By Theorem 79.10.1 we see that $X = U/R$ (quotient in the fppf-topology) is an algebraic space. Using that the smooth topology and the étale topology have the same sheaves (by More on Morphisms, Lemma 37.38.7) we see the map $U \to F$ identifies $F$ as the quotient of $U$ by $R$ for the smooth topology (details omitted). Thus we have morphisms (transformations of functors)

$U \to F \to X.$

By Lemma 79.11.6 we see that $U \to X$ is surjective, flat and locally of finite presentation. By Groupoids in Spaces, Lemma 77.19.5 (and the fact that $j$ is a monomorphism) we have $R = U \times _ X U$. By Descent on Spaces, Lemma 73.11.26 we conclude that $U \to X$ is smooth and surjective (as the projections $R \to U$ are smooth and surjective and $\{ U \to X\}$ is an fppf covering). Hence for any scheme $T$ and morphism $T \to X$ the fibre product $T \times _ X U$ is an algebraic space surjective and smooth over $T$. Choose a scheme $V$ and a surjective étale morphism $V \to T \times _ X U$. Then $\{ V \to T\}$ is a smooth covering such that $V \to T \to X$ lifts to a morphism $V \to U$. This proves that $U \to X$ is surjective as a map of sheaves in the smooth topology. It follows that $F \to X$ is surjective as a map of sheaves in the smooth topology. On the other hand, the map $F \to X$ is injective (as a map of presheaves) since $R = U \times _ X U$. It follows that $F \to X$ is an isomorphism of smooth ($=$ étale) sheaves, see Sites, Lemma 7.11.2 which concludes the proof. $\square$

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